PERMUTATIONS AND COMBINA TIONS

Jacob Bernoulli

(1654-1705)

vEvery body of discovery is mathematical in form because there is no

other guidance we can have – DARWINv

7.1 Introduction

Suppose you have a suitcase with a number lock. The number

lock has 4 wheels each labelled with 10 digits from 0 to 9.

The lock can be opened if 4 specific digits are arranged in a

particular sequence with no repetition. Some how, you have

forgotten this specific sequence of digits. You remember only

the first digit which is 7. In order to open the lock, how

many sequences of 3-digits you may have to check with? To

answer this question, you may, immediately, start listing all

possible arrangements of 9 remaining digits taken 3 at a

time. But, this method will be tedious, because the number

of possible sequences may be large. Here, in this Chapter,

we shall learn some basic counting techniques which will enable us to answer this

question without actually listing 3-digit arrangements. In fact, these techniques will be

useful in determining the number of different ways of arranging and selecting objects

without actually listing them. As a first step, we shall examine a principle which is most

fundamental to the learning of these techniques.

7.2 Fundamental Principle of Counting

Let us consider the following problem. Mohan has 3 pants and 2 shirts. How many

different pairs of a pant and a shirt, can he dress up with? There are 3 ways in which

a pant can be chosen, because there are 3 pants available. Similarly, a shirt can be

chosen in 2 ways. For every choice of a pant, there are 2 choices of a shirt. Therefore,

there are 3 × 2 = 6 pairs of a pant and a shirt.

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PERMUTATIONS AND COMBINATIONS

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PERMUTATIONS AND COMBINATIONS 135

Let us name the three pants as P1, P

2, P

3 and the two shirts as S

1, S

2. Then,

these six possibilities can be illustrated in the Fig. 7.1.

Let us consider another problem

of the same type.

Sabnam has 2 school bags, 3 tiffin boxes

and 2 water bottles. In how many ways

can she carry these items (choosing one

each).

A school bag can be chosen in 2

different ways. After a school bag is

chosen, a tiffin box can be chosen in 3

different ways. Hence, there are

2 × 3 = 6 pairs of school bag and a tiffin

box. For each of these pairs a water

bottle can be chosen in 2 different ways.

Hence, there are 6 × 2 = 12 different ways in which, Sabnam can carry these items to

school. If we name the 2 school bags as B1, B

2, the three tiffin boxes as T

1, T

2, T

3 and

the two water bottles as W1, W

2, these possibilities can be illustrated in the Fig. 7.2.

Fig 7.1

Fig 7.2

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136 MATHEMATICS

In fact, the problems of the above types are solved by applying the following

principle known as the fundamental principle of counting, or, simply, the multiplication

principle, which states that

“If an event can occur in m different ways, following which another event

can occur in n different ways, then the total number of occurrence of the events

in the given order is m×n.”

The above principle can be generalised for any finite number of events. For

example, for 3 events, the principle is as follows:

‘If an event can occur in m different ways, following which another event can

occur in n different ways, following which a third event can occur in p different ways,

then the total number of occurrence to ‘the events in the given order is m × n × p.”

In the first problem, the required number of ways of wearing a pant and a shirt

was the number of different ways of the occurence of the following events in succession:

(i) the event of choosing a pant

(ii) the event of choosing a shirt.

In the second problem, the required number of ways was the number of different

ways of the occurence of the following events in succession:

(i) the event of choosing a school bag

(ii) the event of choosing a tiffin box

(iii) the event of choosing a water bottle.

Here, in both the cases, the events in each problem could occur in various possible

orders. But, we have to choose any one of the possible orders and count the number of

different ways of the occurence of the events in this chosen order.

Example 1 Find the number of 4 letter words, with or without meaning, which can be

formed out of the letters of the word ROSE, where the repetition of the letters is not

allowed.

Solution There are as many words as there are ways of filling in 4 vacant places

by the 4 letters, keeping in mind that the repetition is not allowed. The

first place can be filled in 4 different ways by anyone of the 4 letters R,O,S,E. Following

which, the second place can be filled in by anyone of the remaining 3 letters in 3

different ways, following which the third place can be filled in 2 different ways; following

which, the fourth place can be filled in 1 way. Thus, the number of ways in which the

4 places can be filled, by the multiplication principle, is 4 × 3 × 2 × 1 = 24. Hence, the

required number of words is 24.

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ANote If the repetition of the letters was allowed, how many words can be formed?

One can easily understand that each of the 4 vacant places can be filled in succession

in 4 different ways. Hence, the required number of words = 4 × 4 × 4 × 4 = 256.

Example 2 Given 4 flags of different colours, how many different signals can be

generated, if a signal requires the use of 2 flags one below the other?

Solution There will be as many signals as there are ways of filling in 2 vacant places

in succession by the 4 flags of different colours. The upper vacant place can

be filled in 4 different ways by anyone of the 4 flags; following which, the lower vacant

place can be filled in 3 different ways by anyone of the remaining 3 different flags.

Hence, by the multiplication principle, the required number of signals = 4 × 3 = 12.

Example 3 How many 2 digit even numbers can be formed from the digits

1, 2, 3, 4, 5 if the digits can be repeated?

Solution There will be as many ways as there are ways of filling 2 vacant places

in succession by the five given digits. Here, in this case, we start filling in unit’s

place, because the options for this place are 2 and 4 only and this can be done in 2

ways; following which the ten’s place can be filled by any of the 5 digits in 5 different

ways as the digits can be repeated. Therefore, by the multiplication principle, the required

number of two digits even numbers is 2 × 5, i.e., 10.

Example 4 Find the number of different signals that can be generated by arranging at

least 2 flags in order (one below the other) on a vertical staff, if five different flags are

available.

Solution A signal can consist of either 2 flags, 3 flags, 4 flags or 5 flags. Now, let us

count the possible number of signals consisting of 2 flags, 3 flags, 4 flags and 5 flags

separately and then add the respective numbers.

There will be as many 2 flag signals as there are ways of filling in 2 vacant places

in succession by the 5 flags available. By Multiplication rule, the number of

ways is 5 × 4 = 20.

Similarly, there will be as many 3 flag signals as there are ways of filling in 3

vacant places in succession by the 5 flags.

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138 MATHEMATICS

The number of ways is 5 × 4 × 3 = 60.

Continuing the same way, we find that

The number of 4 flag signals = 5 × 4 × 3 × 2 = 120

and the number of 5 flag signals = 5 × 4 × 3 × 2 × 1 = 120

Therefore, the required no of signals = 20 + 60 + 120 + 120 = 320.

EXERCISE 7.1

1. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5

assuming that

(i) repetition of the digits is allowed?

(ii) repetition of the digits is not allowed?

2. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the

digits can be repeated?

3. How many 4-letter code can be formed using the first 10 letters of the English

alphabet, if no letter can be repeated?

4. How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if

each number starts with 67 and no digit appears more than once?

5. A coin is tossed 3 times and the outcomes are recorded. How many possible

outcomes are there?

6. Given 5 flags of different colours, how many different signals can be generated if

each signal requires the use of 2 flags, one below the other?

7.3 Permutations

In Example 1 of the previous Section, we are actually counting the different possible

arrangements of the letters such as ROSE, REOS, ..., etc. Here, in this list, each

arrangement is different from other. In other words, the order of writing the letters is

important. Each arrangement is called a permutation of 4 different letters taken all

at a time. Now, if we have to determine the number of 3-letter words, with or without

meaning, which can be formed out of the letters of the word NUMBER, where the

repetition of the letters is not allowed, we need to count the arrangements NUM,

NMU, MUN, NUB, ..., etc. Here, we are counting the permutations of 6 different

letters taken 3 at a time. The required number of words = 6 × 5 × 4 = 120 (by using

multiplication principle).

If the repetition of the letters was allowed, the required number of words would

be 6 × 6 × 6 = 216.

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Definition 1 A permutation is an arrangement in a definite order of a number of

objects taken some or all at a time.

In the following sub-section, we shall obtain the formula needed to answer these

questions immediately.

7.3.1 Permutations when all the objects are distinct

Theorem 1 The number of permutations of n different objects taken r at a time,

where 0 < r ≤ n and the objects do not repeat is n ( n – 1) ( n – 2). . .( n – r + 1),

which is denoted by nPr.

Proof There will be as many permutations as there are ways of filling in r vacant

places . . . by

← r vacant places →

the n objects. The first place can be filled in n ways; following which, the second place

can be filled in (n – 1) ways, following which the third place can be filled in (n – 2)

ways,..., the rth place can be filled in (n – (r – 1)) ways. Therefore, the number of

ways of filling in r vacant places in succession is n(n – 1) (n – 2) . . . (n – (r – 1)) or

n ( n – 1) (n – 2) ... (n – r + 1)

This expression for nPr is cumbersome and we need a notation which will help to

reduce the size of this expression. The symbol n! (read as factorial n or n factorial )

comes to our rescue. In the following text we will learn what actually n! means.

7.3.2 Factorial notation The notation n! represents the product of first n natural

numbers, i.e., the product 1 × 2 × 3 × . . . × (n – 1) × n is denoted as n!. We read this

symbol as ‘n factorial’. Thus, 1 × 2 × 3 × 4 . . . × (n – 1) × n = n !

1 = 1 !

1 × 2 = 2 !

1× 2 × 3 = 3 !

1 × 2 × 3 × 4 = 4 ! and so on.

We define 0 ! = 1

We can write 5 ! = 5 × 4 ! = 5 × 4 × 3 ! = 5 × 4 × 3 × 2 !

= 5 × 4 × 3 × 2 × 1!

Clearly, for a natural number n

n ! = n (n – 1) !

= n (n – 1) (n – 2) ! [provided (n ≥ 2)]

= n (n – 1) (n – 2) (n – 3) ! [provided (n ≥ 3)]

and so on.

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140 MATHEMATICS

Example 5 Evaluate (i) 5 ! (ii) 7 ! (iii) 7 ! – 5!

Solution (i) 5 ! = 1 × 2 × 3 × 4 × 5 = 120

(ii) 7 ! = 1 × 2 × 3 × 4 × 5 × 6 ×7 = 5040

and (iii) 7 ! – 5! = 5040 – 120 = 4920.

Example 6 Compute (i) 7!

5!(ii) ( )

12!

10! (2!)

Solution (i) We have 7!

5! =

7 6 5!

5!

× × = 7 × 6 = 42

and (ii) ( ) ( )12!

10! 2! =

( )( ) ( )

12 11 10!

10! 2

× ××

= 6 × 11 = 66.

Example 7 Evaluate ( )!

! !

n

r n r− , when n = 5, r = 2.

Solution We have to evaluate ( )5!

2! 5 2 !− (since n = 5, r = 2)

We have ( )5!

2 ! 5 2 !− = 5! 5 4

102! 3! 2

×= =

× .

Example 8 If 1 1

8! 9! 10!

x+ = , find x.

Solution We have 1 1

8! 9 8! 10 9 8!

x+ =

× × ×

Therefore1

19 10 9

x+ =

× or

10

9 10 9

x=

×So x = 100.

EXERCISE 7.2

1. Evaluate

(i) 8 ! (ii) 4 ! – 3 !

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2. Is 3 ! + 4 ! = 7 ! ? 3. Compute 8!

6! 2!× 4. If 1 1

6! 7! 8!

x+ = , find x

5. Evaluate ( )!

!

n

n r− , when

(i) n = 6, r = 2 (ii) n = 9, r = 5.

7.3.3 Derivation of the formula for nPr

( )!

P!

nr

n

n r−= , 0 ≤ r ≤ n

Let us now go back to the stage where we had determined the following formula:

nPr = n (n – 1) (n – 2) . . . (n – r + 1)

Multiplying numerator and denomirator by (n – r) (n – r – 1) . . . 3 × 2 × 1, we get

( ) ( ) ( )( )( )( )( )

1 2 1 1 3 2 1P

1 3 2 1

nr

n n n ... n r n r n r ...

n r n r ...

− − − + − − − × ×=

− − − × × = ( )!

!

n

n r− ,

Thus ( )!

P!

nr

n

n r=

− , where 0 < r ≤ n

This is a much more convenient expression for nPr than the previous one.

In particular, when r = n, !

P !0!

nn

nn= =

Counting permutations is merely counting the number of ways in which some or

all objects at a time are rearranged. Arranging no object at all is the same as leaving

behind all the objects and we know that there is only one way of doing so. Thus, we

can have

n P0 = 1 =

! !

! ( 0)!=

−n n

n n... (1)

Therefore, the formula (1) is applicable for r = 0 also.

Thus ( )!

P 0!

nr

n, r n

n r= ≤ ≤

− .

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142 MATHEMATICS

Theorem 2 The number of permutations of n different objects taken r at a time,

where repetition is allowed, is nr.

Proof is very similar to that of Theorem 1 and is left for the reader to arrive at.

Here, we are solving some of the problems of the pervious Section using the

formula for nPr to illustrate its usefulness.

In Example 1, the required number of words = 4P4 = 4! = 24. Here repetition is

not allowed. If repetition is allowed, the required number of words would be 44 = 256.

The number of 3-letter words which can be formed by the letters of the word

NUMBER = 6

3

6!P

3!= = 4 × 5 × 6 = 120. Here, in this case also, the repetition is not

allowed. If the repetition is allowed,the required number of words would be 63 = 216.

The number of ways in which a Chairman and a Vice-Chairman can be chosen

from amongst a group of 12 persons assuming that one person can not hold more than

one position, clearly 12

2

12!P 11 12

10!= = × = 132.

7.3.4 Permutations when all the objects are not distinct objects Suppose we have

to find the number of ways of rearranging the letters of the word ROOT. In this case,

the letters of the word are not all different. There are 2 Os, which are of the same kind.

Let us treat, temporarily, the 2 Os as different, say, O1 and O

2. The number of

permutations of 4-different letters, in this case, taken all at a time

is 4!. Consider one of these permutations say, RO1O

2T. Corresponding to this

permutation,we have 2 ! permutations RO1O

2T and RO

2O

1T which will be exactly the

same permutation if O1 and O

2 are not treated as different, i.e., if O

1 and O

2 are the

same O at both places.

Therefore, the required number of permutations = 4!

3 4 122!

= × = .

Permutations when O1, O

2 are Permutations when O

1, O

2 are

different. the same O.

1 2

2 1

RO O T

RO O T

R O O T

1 2

2 1

T O O R

T O O R

T O O R

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1 2

2 1

R O T O

R O T O

R O T O

1 2

2 1

T O R O

T O R O

T O R O

1 2

2 1

R T O O

R T O O

R T O O

1 2

2 1

T R O O

T R O O

T R O O

1 2

2 1

O O R T

O O T R

O O R T

1 2

2 1

O R O T

O R O T

O R O T

1 2

2 1

O T O R

O T O R

O T O R

1 2

2 1

O R T O

O R T O

O R T O

1 2

2 1

O T R O

O T R O

O T R O

1 2

2 1

O O T R

O O T R

O O T R

Let us now find the number of ways of rearranging the letters of the word

INSTITUTE. In this case there are 9 letters, in which I appears 2 times and T appears

3 times.

Temporarily, let us treat these letters different and name them as I1, I

2, T

1, T

2, T

3.

The number of permutations of 9 different letters, in this case, taken all at a time is 9 !.

Consider one such permutation, say, I1 NT

1 SI

2 T

2 U E T

3. Here if I

1, I

2 are not same

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144 MATHEMATICS

and T1, T

2, T

3 are not same, then I

1, I

2 can be arranged in 2! ways and T

1, T

2, T

3 can

be arranged in 3! ways. Therefore, 2! × 3! permutations will be just the same permutation

corresponding to this chosen permutation I1NT

1SI

2T

2UET

3. Hence, total number of

different permutations will be 9!

2! 3!

We can state (without proof) the following theorems:

Theorem 3 The number of permutations of n objects, where p objects are of the

same kind and rest are all different = !

!

n

p.

In fact, we have a more general theorem.

Theorem 4 The number of permutations of n objects, where p1 objects are of one

kind, p2 are of second kind, ..., p

k are of kth kind and the rest, if any, are of different

kind is 1 2

!

! ! !k

n

p p ... p.

Example 9 Find the number of permutations of the letters of the word ALLAHABAD.

Solution Here, there are 9 objects (letters) of which there are 4A’s, 2 L’s and rest are

all different.

Therefore, the required number of arrangements = 9! 5 6 7 8 9

4!2! 2

× × × ×= = 7560

Example 10 How many 4-digit numbers can be formed by using the digits 1 to 9 if

repetition of digits is not allowed?

Solution Here order matters for example 1234 and 1324 are two different numbers.

Therefore, there will be as many 4 digit numbers as there are permutations of 9 different

digits taken 4 at a time.

Therefore, the required 4 digit numbers ( )9

4

9! 9!= P = =

9 – 4 ! 5! = 9 × 8 × 7 × 6 = 3024.

Example 11 How many numbers lying between 100 and 1000 can be formed with the

digits 0, 1, 2, 3, 4, 5, if the repetition of the digits is not allowed?

Solution Every number between 100 and 1000 is a 3-digit number. We, first, have to

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count the permutations of 6 digits taken 3 at a time. This number would be 6P3. But,

these permutations will include those also where 0 is at the 100’s place. For example,

092, 042, . . ., etc are such numbers which are actually 2-digit numbers and hence the

number of such numbers has to be subtracted from 6P3 to get the required number. To

get the number of such numbers, we fix 0 at the 100’s place and rearrange the remaining

5 digits taking 2 at a time. This number is 5P2. So

The required number 6 53 2

6! 5!= P P

3! 3!− = −

= 4 × 5 × 6 – 4 ×5 = 100

Example 12 Find the value of n such that

(i)5 3P 42 P 4n n , n= > (ii)

4

–14

P 5=

3P

n

n , n > 4

Solution (i) Given that

5 3P 42 Pn n=or n (n – 1) (n – 2) (n – 3) (n – 4) = 42 n(n – 1) (n – 2)

Since n > 4 so n(n – 1) (n – 2) ≠ 0

Therefore, by dividing both sides by n(n – 1) (n – 2), we get

(n – 3 (n – 4) = 42

or n2 – 7n – 30 = 0

or n2 – 10n + 3n – 30

or (n – 10) (n + 3) = 0

or n – 10 = 0 or n + 3 = 0

or n = 10 or n = – 3

As n cannot be negative, so n = 10.

(ii) Given that 4

–14

P 5

3P

n

n=

Therefore 3n (n – 1) (n – 2) (n – 3) = 5(n – 1) (n – 2) (n – 3) (n – 4)

or 3n = 5 (n – 4) [as (n – 1) (n – 2) (n – 3) ≠ 0, n > 4]

or n = 10.

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Example 13 Find r, if 5 4Pr = 6 5P

r–1 .

Solution We have 4 515 P 6 Pr r −=

or ( ) ( )4! 5!

5 64 ! 5 1 !r r

× = ×− − +

or ( ) ( ) ( )( )5! 6 5!

4 ! 5 1 5 5 1 !r r r r

×=

− − + − − −

or (6 – r) (5 – r) = 6

or r2 – 11r + 24 = 0

or r2 – 8r – 3r + 24 = 0

or (r – 8) (r – 3) = 0

or r = 8 or r = 3.

Hence r = 8, 3.

Example 14 Find the number of different 8-letter arrangements that can be made

from the letters of the word DAUGHTER so that

(i) all vowels occur together (ii) all vowels do not occur together.

Solution (i) There are 8 different letters in the word DAUGHTER, in which there

are 3 vowels, namely, A, U and E. Since the vowels have to occur together, we can for

the time being, assume them as a single object (AUE). This single object together with

5 remaining letters (objects) will be counted as 6 objects. Then we count permutations

of these 6 objects taken all at a time. This number would be 6P6 = 6!. Corresponding to

each of these permutations, we shall have 3! permutations of the three vowels A, U, E

taken all at a time . Hence, by the multiplication principle the required number of

permutations = 6 ! × 3 ! = 4320.

(ii) If we have to count those permutations in which all vowels are never

together, we first have to find all possible arrangments of 8 letters taken all at a time,

which can be done in 8! ways. Then, we have to subtract from this number, the number

of permutations in which the vowels are always together.

Therefore, the required number 8 ! – 6 ! × 3 ! = 6 ! (7×8 – 6)

= 2 × 6 ! (28 – 3)

= 50 × 6 ! = 50 × 720 = 36000

Example 15 In how many ways can 4 red, 3 yellow and 2 green discs be arranged in

a row if the discs of the same colour are indistinguishable ?

Solution Total number of discs are 4 + 3 + 2 = 9. Out of 9 discs, 4 are of the first kind

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(red), 3 are of the second kind (yellow) and 2 are of the third kind (green).

Therefore, the number of arrangements 9!

=12604! 3! 2!

.

Example 16 Find the number of arrangements of the letters of the word

INDEPENDENCE. In how many of these arrangements,

(i) do the words start with P

(ii) do all the vowels always occur together

(iii) do the vowels never occur together

(iv) do the words begin with I and end in P?

Solution There are 12 letters, of which N appears 3 times, E appears 4 times and D

appears 2 times and the rest are all different. Therefore

The required number of arrangements 12!

16632003! 4! 2!

= =

(i) Let us fix P at the extreme left position, we, then, count the arrangements of the

remaining 11 letters. Therefore, the required number of words starting with P

11!138600

3! 2! 4!= = .

(ii) There are 5 vowels in the given word, which are 4 Es and 1 I. Since, they have

to always occur together, we treat them as a single object EEEEI for the time

being. This single object together with 7 remaining objects will account for 8

objects. These 8 objects, in which there are 3Ns and 2 Ds, can be rearranged in

8!

3! 2! ways. Corresponding to each of these arrangements, the 5 vowels E, E, E,

E and I can be rearranged in 5!

4! ways. Therefore, by multiplication principle,

the required number of arrangements

8! 5!= 16800

3! 2! 4!× =

(iii) The required number of arrangements

= the total number of arrangements (without any restriction) – the number

of arrangements where all the vowels occur together.

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148 MATHEMATICS

= 1663200 – 16800 = 1646400

(iv) Let us fix I and P at the extreme ends (I at the left end and P at the right end).

We are left with 10 letters.

Hence, the required number of arrangements

= 10!

3! 2! 4!= 12600

EXERCISE 7.3

1. How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is

repeated?

2. How many 4-digit numbers are there with no digit repeated?

3. How many 3-digit even numbers can be made using the digits

1, 2, 3, 4, 6, 7, if no digit is repeated?

4. Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4,

5 if no digit is repeated. How many of these will be even?

5. From a committee of 8 persons, in how many ways can we choose a chairman

and a vice chairman assuming one person can not hold more than one position?

6. Find n if n – 1P3 : nP

4 = 1 : 9.

7. Find r if (i) 5 61P 2 Pr r−= (ii) 5 6

1P Pr r−= .

8. How many words, with or without meaning, can be formed using all the letters of

the word EQUATION, using each letter exactly once?

9. How many words, with or without meaning can be made from the letters of the

word MONDAY, assuming that no letter is repeated, if.

(i) 4 letters are used at a time, (ii) all letters are used at a time,

(iii) all letters are used but first letter is a vowel?

10. In how many of the distinct permutations of the letters in MISSISSIPPI do the

four I’s not come together?

11. In how many ways can the letters of the word PERMUTATIONS be arranged if the

(i) words start with P and end with S, (ii) vowels are all together,

(iii) there are always 4 letters between P and S?

7.4 Combinations

Let us now assume that there is a group of 3 lawn tennis players X, Y, Z. A team

consisting of 2 players is to be formed. In how many ways can we do so? Is the team

of X and Y different from the team of Y and X ? Here, order is not important.

In fact, there are only 3 possible ways in which the team could be constructed.

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These are XY, YZ and ZX (Fig 7.3).

Here, each selection is called a combination of 3 different objects taken 2 at a time.

In a combination, the order is not important.

Now consider some more illustrations.

Twelve persons meet in a room and each shakes hand with all the others. How do

we determine the number of hand shakes. X shaking hands with Y and Y with X will

not be two different hand shakes. Here, order is not important. There will be as many

hand shakes as there are combinations of 12 different things taken 2 at a time.

Seven points lie on a circle. How many chords can be drawn by joining these

points pairwise? There will be as many chords as there are combinations of 7 different

things taken 2 at a time.

Now, we obtain the formula for finding the number of combinations of n different

objects taken r at a time, denoted by nCr..

Suppose we have 4 different objects A, B, C and D. Taking 2 at a time, if we have

to make combinations, these will be AB, AC, AD, BC, BD, CD. Here, AB and BA are

the same combination as order does not alter the combination. This is why we have not

included BA, CA, DA, CB, DB and DC in this list. There are as many as 6 combinations

of 4 different objects taken 2 at a time, i.e., 4C2 = 6.

Corresponding to each combination in the list, we can arrive at 2! permutations as

2 objects in each combination can be rearranged in 2! ways. Hence, the number of

permutations = 4C2 × 2!.

On the other hand, the number of permutations of 4 different things taken 2 at

a time = 4P2.

Therefore 4P2 = 4C

2 ××××× 2! or ( )

42

4!C

4 2 ! 2!=

−

Now, let us suppose that we have 5 different objects A, B, C, D, E. Taking 3 at a

time, if we have to make combinations, these will be ABC, ABD, ABE, BCD, BCE,

CDE, ACE, ACD, ADE, BDE. Corresponding to each of these 5C3 combinations, there

are 3! permutations, because, the three objects in each combination can be

Fig. 7.3

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150 MATHEMATICS

rearranged in 3 ! ways. Therefore, the total of permutations = 53C 3!×

Therefore 5P3 = 5C

3 × 3! or ( )

53

5!C

5 3 ! 3!=

−

These examples suggest the following theorem showing relationship between

permutaion and combination:

Theorem 5 P C !n nr r r= , 0 < r ≤ n.

Proof Corresponding to each combination of nCr, we have r ! permutations, because

r objects in every combination can be rearranged in r ! ways.

Hence, the total number of permutations of n different things taken r at a time

is nCr × r!. On the other hand, it is P

n

r. Thus

P C !n nr r r= × , 0 r n< ≤ .

Remarks 1. From above ( )!

C !!

nr

nr

n r= ×

− , i.e., ( )!

C! !

nr

n

r n r=

− .

In particular, if r n= , !

C 1! 0!

nn

n

n= = .

2. We define nC0 = 1, i.e., the number of combinations of n different things taken

nothing at all is considered to be 1. Counting combinations is merely counting the

number of ways in which some or all objects at a time are selected. Selecting

nothing at all is the same as leaving behind all the objects and we know that there

is only one way of doing so. This way we define nC0 = 1.

3. As ( ) 0

!1 C

0! 0 !

nn

n= =

−, the formula ( )

!C

! !

n

r

n

r n r

=−

is applicable for r = 0 also.

Hence

( )!

C! !

n

r

n

r n r

=− , 0 ≤ r ≤ n.

4. ( ) ( )( )!

C! !

nn r

n

n r n n r− =

− − − = ( )

!

! !

n

n r r− = Cnr,

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i.e., selecting r objects out of n objects is same as rejecting (n – r) objects.

5. nCa = nC

b ⇒ a = b or a = n – b, i.e., n = a + b

Theorem 6 11C C Cn n n

r r r+

−+ =

Proof We have ( ) ( ) ( )1

! !C C

! ! 1 ! 1 !

n nr r

n n

r n r r n r−+ = +

− − − +

= ( ) ( )!

1 ! !

n

r r n r× − − + ( ) ( ) ( )

!

1 ! 1 !

n

r n r n r− − + −

= ( ) ( )!

1 ! !

n

r n r− − 1 1

1r n r

+ − +

= ( ) ( ) ( )! 1

1 ! ! 1

n n r r

r n r r n r

− + +×

− − − + = ( )

( )11 !

C! 1 !

nr

n

r n r

++=

+ −

Example 17 If 9 8C Cn n= , find

17C

n .

Solution We have 9 8C Cn n=

i.e., ( ) ( )! !

9! 9 ! 8 ! 8!

n n

n n=

− −

or1 1

9 8n=

− or n – 8 = 9 or n = 17

Therefore 1717 17C C 1n = = .

Example 18 A committee of 3 persons is to be constituted from a group of 2 men and

3 women. In how many ways can this be done? How many of these committees would

consist of 1 man and 2 women?

Solution Here, order does not matter. Therefore, we need to count combinations.

There will be as many committees as there are combinations of 5 different persons

taken 3 at a time. Hence, the required number of ways = 5

3

5! 4 5C 10

3! 2! 2

×= = = .

Now, 1 man can be selected from 2 men in 2C1 ways and 2 women can be

selected from 3 women in 3C2 ways. Therefore, the required number of committees

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152 MATHEMATICS

= 2 3

1 2

2! 3!C C 6

1! 1! 2! 1!× = × = .

Example 19 What is the number of ways of choosing 4 cards from a pack of 52

playing cards? In how many of these

(i) four cards are of the same suit,

(ii) four cards belong to four different suits,

(iii) are face cards,

(iv) two are red cards and two are black cards,

(v) cards are of the same colour?

Solution There will be as many ways of choosing 4 cards from 52 cards as there are

combinations of 52 different things, taken 4 at a time. Therefore

The required number of ways = 52

4

52! 49 50 51 52C

4! 48! 2 3 4

× × ×= =

× ×

= 270725

(i) There are four suits: diamond, club, spade, heart and there are 13 cards of each

suit. Therefore, there are 13C4 ways of choosing 4 diamonds. Similarly, there are

13C4 ways of choosing 4 clubs, 13C

4 ways of choosing 4 spades and 13C

4 ways of

choosing 4 hearts. Therefore

The required number of ways = 13C4 + 13C

4 + 13C

4 + 13C

4.

=13!

4 28604! 9!

× =

(ii) There are13 cards in each suit.

Therefore, there are 13C1

ways of choosing 1 card from 13 cards of diamond,13C

1 ways of choosing 1 card from 13 cards of hearts, 13C

1 ways of choosing 1

card from 13 cards of clubs, 13C1

ways of choosing 1 card from 13 cards of

spades. Hence, by multiplication principle, the required number of ways

= 13C1

× 13C

1 ×

13C

1×

13C

1 = 134

(iii) There are 12 face cards and 4 are to be selected out of these 12 cards. This can be

done in 12C4 ways. Therefore, the required number of ways =

12!495

4! 8!= .

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(iv) There are 26 red cards and 26 black cards. Therefore, the required number of

ways = 26C2 ×

26C2

= ( )2

226!325

2! 24!

=

= 105625

(v) 4 red cards can be selected out of 26 red cards in 26C4 ways.

4 black cards can be selected out of 26 black cards in 26C4ways.

Therefore, the required number of ways = 26C4 + 26C

4

= 26!

24! 22!

× = 29900.

EXERCISE 7.4

1. If nC8 = nC

2, find nC

2.

2. Determine n if

(i) 2nC3 : nC

3 = 12 : 1 (ii) 2nC

3 : nC

3 = 11 : 1

3. How many chords can be drawn through 21 points on a circle?

4. In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and

4 girls?

5. Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5

blue balls if each selection consists of 3 balls of each colour.

6. Determine the number of 5 card combinations out of a deck of 52 cards if there

is exactly one ace in each combination.

7. In how many ways can one select a cricket team of eleven from 17 players in

which only 5 players can bowl if each cricket team of 11 must include exactly 4

bowlers?

8. A bag contains 5 black and 6 red balls. Determine the number of ways in which

2 black and 3 red balls can be selected.

9. In how many ways can a student choose a programme of 5 courses if 9 courses

are available and 2 specific courses are compulsory for every student?

Miscellaneous Examples

Example 20 How many words, with or without meaning, each of 3 vowels and 2

consonants can be formed from the letters of the word INVOLUTE ?

Solution In the word INVOLUTE, there are 4 vowels, namely, I,O,E,Uand 4

consonants, namely, N, V, L and T.

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154 MATHEMATICS

The number of ways of selecting 3 vowels out of 4 = 4C3 = 4.

The number of ways of selecting 2 consonants out of 4 = 4C2 = 6.

Therefore, the number of combinations of 3 vowels and 2 consonants is

4 × 6 = 24.

Now, each of these 24 combinations has 5 letters which can be arranged among

themselves in 5 ! ways. Therefore, the required number of different words is

24 × 5 ! = 2880.

Example 21 A group consists of 4 girls and 7 boys. In how many ways can a team of

5 members be selected if the team has (i) no girl ? (ii) at least one boy and one girl ?

(iii) at least 3 girls ?

Solution (i) Since, the team will not include any girl, therefore, only boys are to be

selected. 5 boys out of 7 boys can be selected in 7C5 ways. Therefore, the required

number of ways = 7

5

7! 6 7C 21

5! 2! 2

×= = =

(ii) Since, at least one boy and one girl are to be there in every team. Therefore, the

team can consist of

(a) 1 boy and 4 girls (b) 2 boys and 3 girls

(c) 3 boys and 2 girls (d) 4 boys and 1 girl.

1 boy and 4 girls can be selected in 7C1 × 4C

4 ways.

2 boys and 3 girls can be selected in 7C2 × 4C

3 ways.

3 boys and 2 girls can be selected in 7C3 × 4C

2 ways.

4 boys and 1 girl can be selected in 7C4 × 4C

1 ways.

Therefore, the required number of ways

= 7C1 × 4C

4 + 7C

2 × 4C

3 +

7C3 × 4C

2 + 7C

4 × 4C

1

= 7 + 84 + 210 + 140 = 441

(iii) Since, the team has to consist of at least 3 girls, the team can consist of

(a) 3 girls and 2 boys, or (b) 4 girls and 1 boy.

Note that the team cannot have all 5 girls, because, the group has only 4 girls.

3 girls and 2 boys can be selected in 4C3 × 7C

2 ways.

4 girls and 1 boy can be selected in 4C4 × 7C

1 ways.

Therefore, the required number of ways

= 4C3 × 7C

2 + 4C

4 × 7C

1 = 84 + 7 = 91

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Example 22 Find the number of words with or without meaning which can be made

using all the letters of the word AGAIN. If these words are written as in a dictionary,

what will be the 50th word?

Solution There are 5 letters in the word AGAIN, in which A appears 2 times. Therefore,

the required number of words = 5!

602!

= .

To get the number of words starting with A, we fix the letter A at the extreme left

position, we then rearrange the remaining 4 letters taken all at a time. There will be as

many arrangements of these 4 letters taken 4 at a time as there are permutations of 4

different things taken 4 at a time. Hence, the number of words starting with

A = 4! = 24. Then, starting with G, the number of words 4!

2!= = 12 as after placing G

at the extreme left position, we are left with the letters A, A, I and N. Similarly, there

are 12 words starting with the next letter I. Total number of words so far obtained

= 24 + 12 + 12 =48.

The 49th word is NAAGI. The 50th word is NAAIG.

Example 23 How many numbers greater than 1000000 can be formed by using the

digits 1, 2, 0, 2, 4, 2, 4?

Solution Since, 1000000 is a 7-digit number and the number of digits to be used is also

7. Therefore, the numbers to be counted will be 7-digit only. Also, the numbers have to

be greater than 1000000, so they can begin either with 1, 2 or 4.

The number of numbers beginning with 1 = 6! 4 5 6

3! 2! 2

× ×= = 60, as when 1 is

fixed at the extreme left position, the remaining digits to be rearranged will be 0, 2, 2, 2,

4, 4, in which there are 3, 2s and 2, 4s.

Total numbers begining with 2

= 6! 3 4 5 6

2! 2! 2

× × ×= = 180

and total numbers begining with 4 6!

4 5 63!

= = × × = 120

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156 MATHEMATICS

Therefore, the required number of numbers = 60 + 180 + 120 = 360.

Alternative Method

The number of 7-digit arrangements, clearly, 7!

4203! 2!

= . But, this will include those

numbers also, which have 0 at the extreme left position. The number of such

arrangements 6!

3! 2! (by fixing 0 at the extreme left position) = 60.

Therefore, the required number of numbers = 420 – 60 = 360.

ANote If one or more than one digits given in the list is repeated, it will be

understood that in any number, the digits can be used as many times as is given in

the list, e.g., in the above example 1 and 0 can be used only once whereas 2 and 4

can be used 3 times and 2 times, respectively.

Example 24 In how many ways can 5 girls and 3 boys be seated in a row so that no

two boys are together?

Solution Let us first seat the 5 girls. This can be done in 5! ways. For each such

arrangement, the three boys can be seated only at the cross marked places.

× G × G × G × G × G ×.

There are 6 cross marked places and the three boys can be seated in 6P3 ways.

Hence, by multiplication principle, the total number of ways

= 5! × 6P3 =

6!5!×

3!

= 4 × 5 × 2 × 3 × 4 × 5 × 6 = 14400.

Miscellaneous Exercise on Chapter 7

1. How many words, with or without meaning, each of 2 vowels and 3 consonants

can be formed from the letters of the word DAUGHTER ?

2. How many words, with or without meaning, can be formed using all the letters of

the word EQUATION at a time so that the vowels and consonants occur together?

3. A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways

can this be done when the committee consists of:

(i) exactly 3 girls ? (ii) atleast 3 girls ? (iii) atmost 3 girls ?

4. If the different permutations of all the letter of the word EXAMINATION are

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listed as in a dictionary, how many words are there in this list before the first

word starting with E ?

5. How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9

which are divisible by 10 and no digit is repeated ?

6. The English alphabet has 5 vowels and 21 consonants. How many words with

two different vowels and 2 different consonants can be formed from the

alphabet ?

7. In an examination, a question paper consists of 12 questions divided into two

parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student

is required to attempt 8 questions in all, selecting at least 3 from each part. In

how many ways can a student select the questions ?

8. Determine the number of 5-card combinations out of a deck of 52 cards if each

selection of 5 cards has exactly one king.

9. It is required to seat 5 men and 4 women in a row so that the women occupy the

even places. How many such arrangements are possible ?

10. From a class of 25 students, 10 are to be chosen for an excursion party. There

are 3 students who decide that either all of them will join or none of them will

join. In how many ways can the excursion party be chosen ?

11. In how many ways can the letters of the word ASSASSINATION be arranged

so that all the S’s are together ?

Summary

®Fundamental principle of counting If an event can occur in m different

ways, following which another event can occur in n different ways, then the

total number of occurrence of the events in the given order is m × n.

®The number of permutations of n different things taken r at a time, where

repetition is not allowed, is denoted by nPr and is given by nP

r =

!

( )!

n

n r− ,

where 0 ≤ r ≤ n.

®n! = 1 × 2 × 3 × ...×n

®n! = n × (n – 1) !

®The number of permutations of n different things, taken r at a time, where

repeatition is allowed, is nr.

®The number of permutations of n objects taken all at a time, where p1 objects

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158 MATHEMATICS

are of first kind, p2 objects are of the second kind, ..., p

k objects are of the kth

kind and rest, if any, are all different is 1 2

!

! ! !k

n

p p ... p .

®The number of combinations of n different things taken r at a time, denoted by

nCr , is given by nC

r =

!

! !

n

r ( n r )=

− , 0 ≤ r ≤ n.

Historical Note

The concepts of permutations and combinations can be traced back to the advent

of Jainism in India and perhaps even earlier. The credit, however, goes to the

Jains who treated its subject matter as a self-contained topic in mathematics,

under the name Vikalpa.

Among the Jains, Mahavira, (around 850) is perhaps the world’s first

mathematician credited with providing the general formulae for permutations and

combinations.

In the 6th century B.C., Sushruta, in his medicinal work, Sushruta Samhita,

asserts that 63 combinations can be made out of 6 different tastes, taken one at a

time, two at a time, etc. Pingala, a Sanskrit scholar around third century B.C.,

gives the method of determining the number of combinations of a given number

of letters, taken one at a time, two at a time, etc. in his work Chhanda Sutra.

Bhaskaracharya (born 1114) treated the subject matter of permutations and

combinations under the name Anka Pasha in his famous work Lilavati. In addition

to the general formulae for nCr and nP

r already provided by Mahavira,

Bhaskaracharya gives several important theorems and results concerning the

subject.

Outside India, the subject matter of permutations and combinations had its

humble beginnings in China in the famous book I–King (Book of changes). It is

difficult to give the approximate time of this work, since in 213 B.C., the emperor

had ordered all books and manuscripts in the country to be burnt which fortunately

was not completely carried out. Greeks and later Latin writers also did some

scattered work on the theory of permutations and combinations.

Some Arabic and Hebrew writers used the concepts of permutations and

combinations in studying astronomy. Rabbi ben Ezra, for instance, determined

the number of combinations of known planets taken two at a time, three at a time

and so on. This was around 1140. It appears that Rabbi ben Ezra did not know

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the formula for nCr. However, he was aware that nC

r = nC

n–r for specific values

n and r. In 1321, Levi Ben Gerson, another Hebrew writer came up with the

formulae for nPr , nP

n and the general formula for nC

r.

The first book which gives a complete treatment of the subject matter of

permutations and combinations is Ars Conjectandi written by a Swiss, Jacob

Bernoulli (1654 – 1705), posthumously published in 1713. This book contains

essentially the theory of permutations and combinations as is known today.

— vvvvv —

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PERMUTATIONS AND COMBINA TIONS